# An Experimental and Numerical Study of *Calliphora* Wing Structure

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s11340-009-9316-8

- Cite this article as:
- Ganguli, R., Gorb, S., Lehmann, F. et al. Exp Mech (2010) 50: 1183. doi:10.1007/s11340-009-9316-8

- 13 Citations
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## Abstract

Experiments are performed to determine the mass and stiffness variations along the wing of the blowfly *Calliphora*. The results are obtained for a pairs of wings of 10 male flies and fresh wings are used. The wing is divided into nine locations along the span and seven locations along the chord based on venation patterns. The length and mass of the sections is measured and the mass per unit length is calculated. The bending stiffness measurements are taken at three locations, basal (near root), medial and distal (near tip) of the fly wing. Torsional stiffness measurements are also made and the elastic axis of the wing is approximately located. The experimental data is then used for structural modeling of the wing as a stepped cantilever beam with nine spanwise sections of varying mass per unit lengths, flexural rigidity (*EI*) and torsional rigidity (*GJ*) values. Inertial values of nine sections are found to approximately vary according to an exponentially decreasing law over the nine sections from root to tip and it is used to calculate an approximate value of Young’s modulus of the wing biomaterial. Shear modulus is obtained assuming the wing biomaterial to be isotropic. Natural frequencies, both in bending and torsion, are obtained by solving the homogeneous part of the respective governing differential equations using the finite element method. The results provide a complete analysis of *Calliphora* wing structure and also provide guidelines for the biomimetic structural design of insect-scale flapping wings.

### Keywords

Micro air vehicle*Calliphora*Mass per unit lengthFlexural rigidityTorsional rigidityNatural frequencyFinite element method

**Nomenclature**

*A*area

*b*width

*E*material stiffness (Young’s modulus)

*EI*flexural rigidity

*f*force per unit length

*f*_{0}torque per unit length

*F*applied force

*G*shear modulus

*GJ*torsional rigidity

*h*thickness

*i*imaginary unit

*I*moment of inertia

*I*_{0}mass polar moment of inertia per unit length

*J*polar moment of inertia

*k*bending stiffness

*K*global stiffness matrix

*l*mean section length

*m*mass per unit length

- \(\bar{m}\)
average mass of each segments

*M*global mass matrix

*Q*nodal displacement vectors

*t*time

*x*distance

*y*deflection in y-direction

*y*_{tip}deflection at the tip

*ρ*density

*σ*Poisson’s ratio

*ω*natural frequency

## Introduction

There is a world wide interest in the development and further research on a family of very small size air vehicles having a maximum dimension of 15 cm and a gross weight of 100 g known as Micro Air Vehicles (MAVs). The three main approaches for providing lift for such vehicles are through fixed, rotating and flapping wings. Fixed wings lack low speed flight capability which may be critical for MAVs in indoor situations. Rotary wing can lead to significant noise signatures. Nature provides numerous flying birds and insects which use the flapping wing technology [1, 2]. In fact, one can consider birds and insects to be naturally designed flapping wing micro air vehicles. Therefore, it is of great interest to study natural flyers for insights into biomimetic MAVs [3, 4].

The main mission of MAVs is flying in confined spaces inside buildings, shafts, tunnels, machine rooms etc. This requires power-efficient, highly maneuverable and low speed flight. Such performance is routinely exhibited by flying insects. In particular, flying insects have fascinated MAV researchers because of their excellent flying characteristics. A detailed discussion of the future utility of MAVs and the advantages of considering insect-like flapping wing propulsion has been presented by some researchers [5–7]. Research has also been done on the propulsion and aerodynamics of flapping wings [8–11]. The objective of much of this work is to understand the fundamental physics behind insect flight which typically occurs at Reynolds number of 10 − 10^{4}. Another objective is to develop analytical and numerical models which can be used for the design of insect like MAVs. An important feature of flapping flight aerodynamics is their unsteadiness and formation of a leading edge vortex on the wing. These effects are in addition to the conventional wake shed from the trailing edge of the wing. The modeling effort is therefore complicated. A recent and comprehensive review of insect flight aerodynamics is provided by Wang [12].

Relative to the large number of works on insect flight aerodynamics, very few researchers have focused on insect wing structure. In fact, biological structures are complex composite structures and show very good mechanical properties even though they are made from weak materials [13]. In general, insect wing structures are composed of membranes strengthened by veins. Wang et al. [14] found that the dragonfly veins are complex microstructures consisting of chitin shell, muscle and fibrils. The vein structures are such that they allow the insect to sustain high levels of bending and torsion loads. In another work, Machida et al. [15] found that the costal vein plays an important role in the bending and torsion of the dragonfly wing.

Since the insect wing is a complex mechanical structure, some researchers have modeled it using the finite element method. For example, Smith [16] modeled the veins as three dimensional tubular beam elements of varying thickness for the hawkmoth *Manduca Sexta*. Wootton et al. [17] modeled the wing membrane using orthotropic plane stress quadrilateral (or triangular) elements of varying thickness. Wooton et al. [17] also summarize the current work on the structural modeling of insect wings and show how research has progressed from simple conceptual models of the wing structure to analytical methods and numerical approaches based on the finite element method. A limitation of some of these works is that detailed information about the material and geometric properties of the veins and membranes along the complete wing are required for developing a finite element model.

Another approach for modeling the wing mechanics is to obtain equivalent beam-type variation of the overall stiffness properties experimentally. Taking this approach of experimental work, Combes and Daniel [18, 19] estimated the distribution of flexural stiffness in the dragonfly *Aeshna Multicolor* and the hawkmoth *Manduca Sexta* and approximated the variation as an exponential decline from the wing root to tip. They attached the fresh wings at the root using wax and applied loads at selected sections of the wings to obtain the stiffness properties. They found that the different insects studied followed the general pattern of high stiffness near the wing root and low stiffness near the tip. In addition, they performed finite element simulations using the experimental data and showed considerable levels of wing deflection resulting from distributed lifting forces acting on the wing and needed to support the insect weight. However, Combes and Daniel [18, 19] did not investigate the mass variation along the wing. The variation of mass per unit length along with wing is needed for structural dynamic analysis. As we will see later in this paper, the governing partial differential equations for bending and torsion of beams require mass per unit length and torsional inertia distribution, respectively. While they measured the out-of-plane and in-plane stiffness, they did not measure the torsional stiffness. Ennos [20–22] studied the effects of the torsional rigidity of the insect wings on aerodynamic efficiency. Sunada et al. [23] looked at the torsional deformation and stiffness of dragonfly wings. They found that wing corrugations cause an increase in torsional stiffness due to warping effects. It is clear from videos of flying insects that elasticity plays an important role in insect wings. For example, during stroke reversals, torsional stiffness is important. Recently, Rosenfeld and Wereley [24] studied structural parametric stability of flapping insect wing. They developed a linear time-periodic assumed-mode analysis by modeling the wing structure as a thin beam. The effects of normalized cantilever frequency, bending modes, torsion modes, feathering stroke, location of cross sectional center of gravity and the location of feathering axis on the time periodic stability are shown. In general, a study of the structural aspects of biosystems is becoming increasingly important [25, 26].

*Calliphora*, a picture of which is shown in Fig. 1 (figure from

*wikepedia.org*) and subsequently model the wing into a non-uniform cantilever beam. Since flies are very agile fliers, they are a source of inspiration for micromechanical insect design [27, 28]. While most MAV researchers simply want to construct a flapping wing MAV and do not seek significant levels of biomimesis, a close duplication of insect wings may yield advantages over a new design. It can be seen from Fig. 1 that the fly represents a complete miniature aerospace system, with wings for providing lift and controllability, a vision and nervous system for navigation and guidance, and a body as the payload. Therefore, it allows one to take advantage of the evolutionary design by nature which occurred over very large time scales.

## Experimental Data

### Wing Mass Distribution

The mass measurements are performed using a UMX2 weighing scale manufactured by Mettler Toledo and capable of measuring up to 0.1 *μ*g. The insect wings were cut using special razor blades, manufactured by Ted Pella Inc., under a microscope with a magnification of 3000 and each wing segment weight was measured.

*μ*g, or about 0.4% of the insect weight. The 95% confidence interval using the t-distribution is [228.07

*μ*g, 249.33

*μ*g]. Thus the wings perform a very important function of providing lift with a very low weight. However, biological systems show considerable scatter as individual flies have different weights and are at different stages of their development. The wing weights vary from a minimum of 200.3

*μ*g to a maximum of 272.3

*μ*g with a standard deviation of 22.94

*μ*g. Figure 6 shows the mean and the confidence interval for 95% using t-distribution. Again, the higher wing mass near the root can be observed.

### Geometry

The sectional weights do not indicate the true mass per unit length of each section as the different sections have different lengths, as can be seen in Fig. 3. To obtain the mass per unit length, which is the quantity used for engineering beam design, we measured the section lengths of 10 fly wing pairs.

The wings ranged in length from a minimum of 7.46 mm to a maximum of 9.30 mm and the average length is 8.42 mm. The width ranges from a minimum of 3.07 mm to a maximum of 4.038 mm with an average width of 3.61 mm.

### Mass Per Unit Length

*m*

_{i}) at each section is calculated by dividing the average values of the mass (\(\bar{m_i}\)) with the average value of the length (

*l*

_{i}) of the section, which can be written as

### Bending Stiffness Distribution

Minimum and maximum bending stiffness for 10 pairs of wings

| Basal (min, max) | Medial (min, max) | Distal (min, max) |
---|---|---|---|

Bending up (N/m) | 19.097, 64.305 | 0.826, 12.37 | 0.457, 2.736 |

Bending down (N/m) | 12.236, 48.011 | 1.01, 7.695 | 0.529, 4.478 |

### Torsional Stiffness

The torsional stiffness measurements are performed as follows. First, the elastic axis of the wing is determined in an approximate manner following the procedure outlined by Sunada et al. [23]. A needle is used to apply a load at different spanwise locations. At each spanwise location, the point where only bending resulted with the applied load is found. These points are then connected to determine the elastic axis. The elastic axis is found to be close to the leading edge of the wing, as shown in Fig. 3.

## Structural Modeling of *Calliphora* Wing

*A*

_{i}) are obtained by taking the ratio of mass per unit length of the beam sections, as shown in Fig. 13 in discrete form, with the density of the material (

*ρ*) as expressed by the equation

^{3}, as is typical of insect wings [18]. The data showing variation of areas is plotted in Fig. 25. The mean area was found to be about 0.03315 mm

^{2}with a standard deviation of 0.0032 mm

^{2}. The 95% confidence interval using the Student’s t-distribution is [0.0105 mm

^{2}, 0.0557 mm

^{2}].

*μ*m and a minimum of 3.15

*μ*m. The mean was about 12.19

*μ*m. Note that this thickness variation represents an equivalent mathematical model of the insect wing. The actual wing has variable thickness at different places since the veins are much thicker than the membrane. However, the overall effect of the veins is to create a wing structure which has higher flexural stiffness and thickness near the root region, just like an airplane wing.

### Flexural Rigidity and Natural Frequency

*EI*(

*x*) is the flexural rigidity,

*m*(

*x*) is the mass per unit length,

*y*(

*x*,

*t*) is the deflection and

*f*(

*x*,

*t*) is the force per unit length.

^{ − 6}mm

^{4}.

*Calliphora*wing using the integral beam equation given by

*I*(

*x*) represents the second moment of inertia as a continuous function of distance and its mathematical expression is given in equation (6). The exponential fit allows us to calculate this integral analytically. This integral is evaluated thrice for three different cases for bending stiffness measurements at Basal (near Root), Medial and Distal (near Tip) locations yielding three different values of material stiffness (

*E*). The mean of these three values is taken as the Young’s Modulus for the

*Calliphora*wing which comes out to be about 4.69 × 10

^{10}Nm

^{ − 2}which is close to the value 60 GPa used for the finite element modeling of the insect wing [32]. Moreover, Wang et al. [14] experimentally measured the value of the Young’s modulus 60–80 GPa of hindwing of the dragonfly

*Pantala Flewescens*. The insect wing is a very complicated structure and the net result of the veins, membranes and corrugations on the wings is that they are quite stiff. The insect wings have thus been studied structurally by some researchers such as Wooton [17] using finite element models etc. and the fact that they have a high level of resultant stiffness has been shown. The detailed analysis of insect wings have shown the presence of chitin nanofibres the Young’s modulus of which is over 150 GPa. A detailed discussion of the mechanical properties and design aspects of insect cuticles is given by Vincent and Wegst [33]. The importance of nano structures in insect wings is also mentioned by Watson and Watson [34]. While the science behind the high levels of stiffness in insect wings is not yet fully understood, our study has the limited goal of finding the resultant values for a particular insect, the

*Calliphora*.

*E*) obtained with calculated inertial values of each segment, i.e., (

*EI*)

_{i}=

*EI*

_{i}. The calculated data is shown in Fig. 31. Most of the flexural stiffness seems to be at the root of the wing. The mean of the calculated data for flexural stiffness values for all the nine beam segments gives the mean flexural stiffness as 8.92 × 10

^{ − 8}Nm

^{ − 2}. We can see that the wing is quite rigid near the root where higher structural loads are felt and are very flexible in the outbound regions. It is thus a highly flexible load carrying structure.

Natural frequency of the cantilever wing in bending can be obtained from the homogenous part of the equation (4) which cannot be solved exactly for a non-uniform beam. Therefore, we use the finite element method to model the insect wing structure [35].

*M*is the global mass matrix,

*K*is the global stiffness matrix and

*Q*is the vector of nodal degrees of freedom. For the steady state condition, starting from the equilibrium state, we seek a solution of the form

*U*is the vector of nodal amplitudes of vibrations and

*ω*is the natural frequency. Finally, natural frequency is obtained by solving the generalized eigenvalue problem which can be expressed as

Flapping frequency of *Calliphora* wing is 150 Hz [27]. Natural frequency in bending is obtained 475 Hz which is 3.17 times of the flapping frequency.

### Torsional Rigidity and Natural Frequency

*GJ*(

*x*) is the torsional rigidity,

*I*

_{0}(

*x*) is mass polar moment of inertia per unit length,

*f*

_{0}(

*x*,

*t*) is the applied torque per unit length and

*θ*(

*x*,

*t*) is the angle of twist.

^{ − 2}mm

^{4}. Shear modulus (

*G*) for the

*Calliphora*wing is obtained from the following expression [36]

*σ*) is assumed to be 0.3 [23].

*G*), as obtained from equation (13), with calculated polar moment of inertia of each segment, i.e., (

*GJ*)

_{i}=

*GJ*

_{i}. The calculated data is shown in Fig. 33. The mean of the calculated data for torsional rigidity values for all the nine beam segments gives the mean torsional rigidity as 4.39 × 10

^{ − 4}Nm

^{2}. It can be seen from Fig. 33 that value of the torsional rigidity increases towards the root and having highest value at the location 8. This is because of maximum venation density at this location, which resists the torsional deformation.

Natural frequency of the cantilever wing in torsion is obtained by solving the homogenous part of the equation (11) using the finite element method. In this case, same solution procedure is followed as that of solving the governing differential equation of the non-uniform cantilever wing in bending. Natural frequency of the cantilever wing in torsion is 283 Hz which is 1.57 times of the flapping frequency of *Calliphora* wing.

The present structural modeling and analysis of the *Calliphora* wing may be useful for fundamental understanding of insect wing structure and its dynamic behavior. Moreover, the structural model may be used to study several aspects of insect wing structure such as stability [24]. In order to get a deeper understanding of the insect wing structure, the present non-uniform cantilever beam model can be revised by considering a detailed finite element model which in turn can be coupled to aerodynamic model for investigating the aeroelastic properties of the *Calliphora* wing. This we have taken up as a future work.

Some issues related to measurement uncertainty should be pointed out as they propagate into the numerical results. The Biopac load cells have a minimum 50 g full scale measurement. The accuracy of these load cells is no better than 0.1% FS, which translates to 500 μN. Given that the full range of measurements is between 2500 and 5000 μN, there is a lot of uncertainty in load measurements which are only 1% FS and will significantly effect the accuracy of the moduli measurements.

## Conclusions

Experiments were performed and the mass and stiffness variation of the *Calliphora* wing was obtained. It is found that the mass per unit length is higher in the wing root and decreases towards the wing tip. The mass per unit length is highest near the leading edge of the wing and decreases towards the trailing edge. The wing shows a high level of bending stiffness near the root and there is a sharp fall in stiffness towards the tip. The wing also appears to have different bending stiffness in the upward and downward directions. The torsional stiffness is low and the elastic axis lies near the leading edge.

The wing is modeled as a cantilever beam with nine stepped segments along the wing span using experimental data. Variations of width and thickness along the cantilever wing sections are obtained for the numerical study. The inertia of the beam segments are found to decrease exponentially from root to tip and this is used to estimate the Young’s modulus of the *Calliphora* wing. Shear modulus is obtained assuming the wing is made of isotropic material and these values are in turned used for structural modeling. Flexural rigidity (*EI*) and torsional rigidity (*GJ*) are subsequently obtained and they show an increasing tendency towards the root. Natural frequencies, both in bending and torsion, are obtained by solving the homogeneous part of the respective governing differential equations using the finite element method. It is found that natural frequency in bending and torsion are 3.17 and 1.57 times higher than flapping frequency of *Calliphora* wing, respectively. The results provide a complete analysis of *Calliphora* wing structure and also provide guidelines for the biomimetic structural design of insect-scale flapping wings.

## Acknowledgement

The first author thanks the Alexander von Humboldt foundation for providing a fellowship for conducting part of this work.